*Peter Scholze and Jared Weinstein*

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0022
- Subject:
- Mathematics, Geometry / Topology

This chapter discusses vector bundles and G-torsors on the relative Fargues-Fontaine curve. This is in preparation for the examination of moduli spaces of shtukas. Kedlaya-Liu prove two important ...
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This chapter discusses vector bundles and G-torsors on the relative Fargues-Fontaine curve. This is in preparation for the examination of moduli spaces of shtukas. Kedlaya-Liu prove two important foundational theorems about vector bundles on the Fargues-Fontaine curve. The first is the semicontinuity of the Newton polygon. The second theorem of Kedlaya-Liu concerns the open locus where the Newton polygon is constant 0. For the applications to the moduli spaces of shtukas, one needs to generalize the results to the case of G-torsors for a general reductive group G. The chapter then identifies the classification of G-torsors. It also looks at the semicontinuity of the Newton point.Less

This chapter discusses vector bundles and *G*-torsors on the relative Fargues-Fontaine curve. This is in preparation for the examination of moduli spaces of shtukas. Kedlaya-Liu prove two important foundational theorems about vector bundles on the Fargues-Fontaine curve. The first is the semicontinuity of the Newton polygon. The second theorem of Kedlaya-Liu concerns the open locus where the Newton polygon is constant 0. For the applications to the moduli spaces of shtukas, one needs to generalize the results to the case of *G*-torsors for a general reductive group *G*. The chapter then identifies the classification of *G*-torsors. It also looks at the semicontinuity of the Newton point.

*Peter Scholze and Jared Weinstein*

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0013
- Subject:
- Mathematics, Geometry / Topology

This chapter offers a second lecture on one-legged shtukas. It shows that a shtuka over Spa Cb, a priori defined over Y[0,INFINITY) = Spa Ainf REVERSE SOLIDUS {xk, xL}, actually extends to Y = Spa ...
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This chapter offers a second lecture on one-legged shtukas. It shows that a shtuka over Spa Cb, a priori defined over Y[0,INFINITY) = Spa Ainf REVERSE SOLIDUS {xk, xL}, actually extends to Y = Spa Ainf REVERSE SOLIDUS {xk}. In doing so, the chapter considers the theory of φ-modules over the Robba ring, due to Kedlaya. These are in correspondence with vector bundles over the Fargues-Fontaine curve. The chapter then looks at the proposition that the space Y is an adic space. It also sketches a proof that the functor described in Theorem 13.2.1 is fully faithful. This is more general, and works if C is any perfectoid field (not necessarily algebraically closed).Less

This chapter offers a second lecture on one-legged shtukas. It shows that a shtuka over Spa *C*^{b}, a priori defined over *Y*_{[0,INFINITY)} = Spa A_{inf} REVERSE SOLIDUS {*x*_{k}, *x*_{L}}, actually extends to *Y* = Spa A_{inf} REVERSE SOLIDUS {*x*_{k}}. In doing so, the chapter considers the theory of φ-modules over the Robba ring, due to Kedlaya. These are in correspondence with vector bundles over the Fargues-Fontaine curve. The chapter then looks at the proposition that the space *Y* is an adic space. It also sketches a proof that the functor described in Theorem 13.2.1 is fully faithful. This is more general, and works if *C* is any perfectoid field (not necessarily algebraically closed).